
for all \(u\in L^\alpha (E,m)\) .
This stochastic integration is well-defined on the space \(L^\alpha (E,m)\)  [1]}.
In fact, for simple functions \(u(x)=\sum _{j=1}^n c_j\mathbb {1}_{A_j}(x)\) , where \(c_j\in \mathbb {C}\)  and \(A_j\in \mathcal {E}_0\) , it is easily seen that the integral \(I(u)\)  is well-defined.
Moreover, for any function \(u\in L^\alpha (E,m)\)  one can find a sequence of simple functions \(\lbrace u_n\rbrace _{n=1}^\infty \)  which converges to the function \(u\)  almost everywhere on \(E\)  with \(\vert u_n(x)\vert \le v(x)\)  for some function \(v\in L^\alpha (E,m)\)  for all \(n\in \mathbb {N}\)  and \(x\in E\) .
The sequence \(\lbrace I(u_n)\rbrace _{n=1}^\infty \)  converges in probability, and \(I(u)\)  is then defined as its limit.
